During the last few decades, methods have been developed for reconstructing the spatial distribution of an internal property of an object by acquiring multiple projections of that property and then combining them using a reconstruction algorithm. Although there are various applications of these methods, the medical imaging system using x-rays and computed tomography, the CT scanner, is perhaps the best known. The CT scanner typically obtains the distribution of attenuation in a two-dimensional slice of the object by taking projections through 180 degrees around the slice. However, reconstruction methods also can work in three dimensions. For a three-dimensional reconstruction, projections would be taken over a hemisphere.
The earliest and most often used CT algorithm is commonly known as filtered back projection, or simply FBP. A set of parallel x-rays is sent through a selected slice of the object in the plane of the slice and in a given direction. The attenuation as a function of position across the slice is the one-dimensional projection, or projection function, of the slice. Such projections are obtained for many different directions around the slice giving a set of edge-on projections. The resulting projections of the slice are stored in a computer memory. These projection functions are projected back through a numerical array in the same direction in which they were acquired. Before each function is back-projected, it is filtered by convolving it with a 1/r factor. The result of this process is a two-dimensional image of the distribution of the x-ray attenuation within the selected slice.
A second method for reconstruction from projections, called the Fourier method, is based on Fourier transforms and the Projection-Slice Theorem. With this method, the projections of a slice are obtained as described above. Then the projection functions are Fourier transformed and the transforms are placed as a line of numbers into a two-dimensional numerical array here called F-space. For each projection direction, the corresponding line of numbers goes into F-space through the origin and at right angles to the direction of the projection. When F-space is fully populated and the data has been modified or “filtered”, an inverse Fourier transform of the data in F-space is performed in order to produce the desired distribution of the slice.
In this discussion, the term Fourier transform also includes any similar transform that converts one function into a summation of a series of periodic functions. In very general terms, the Fourier transform takes a first function from a first n-dimensional space, alters it, and puts it into a second n-dimensional space as a second function. This can be accomplished by taking the first function element-by-element, altering each element, and then adding each altered element to the other altered elements in the second space. As an example, if the first function is a one-dimensional array of numbers, then for each of the numbers, a sinusoid, a periodic function, is added into a second one-dimensional array. The amplitude of the sinusoid is proportional to the number and the frequency of the sinusoid is proportional to the location of the number in the input array. With Fourier transforms, these sinusoids are added together as an integral part of the transform process.
For two-dimensional reconstructions, the Projection-Slice Theorem states that the Fourier transform of an edge-on projection of the distribution of a property in a two-dimensional slice of an object is the same as a line of data extracted from the two-dimensional Fourier transform of the distribution, said extracted line being through the origin of the transform and perpendicular to the direction of the projection. For three-dimensional reconstructions, the Projection-Slice Theorem states that the Fourier transform of a projection of the distribution of a property in an object is the same as a plane of data extracted from the three-dimensional Fourier transform of the distribution of the property within the object, said extracted plane being through the origin of the transform and perpendicular to the direction of the projection.
With the Fourier method, the Fourier transform of each projection, whether a one-dimensional projection or a two-dimensional projection, is loaded into F-space as prescribed by the Projection-Slice Theorem. Usually enough projection directions are used so that F-space is filled with data. Since all of the transforms go through the origin of F-space, the data is denser there. Some mechanism, such as multiplying the amplitudes by the distance from the origin, is used to compensate for this non-uniform data density. The distribution of the object is obtained by taking the inverse Fourier transform of the data in F-space.
Both of the methods outlined above require that the rays used for each projection be parallel. The earliest CT scanners had a complex mechanical arrangement that translated a single ray resulting from a single source and a single detector across the object. It then rotated the ray to a different orientation and translated the ray across the object again. In this way, it generated a set of projections each obtained with parallel rays. But it is much more efficient to use more than one ray from the x-ray source. Later CT scanners use divergent x-ray beams, called fan-beams, and an arc or ring of detector elements. Since the rays are no longer parallel, more complex reconstruction methods have to be used. Typically, with planar imaging using fan-beams, the attenuation numbers are collected from all of the rays from all of the projections and then resorted, a process called re-binning, into sets that come from parallel rays. This provides parallel-ray projections. Once re-binning is done, the usual FBP or Fourier reconstruction algorithms can be employed.
In order to become even more efficient, multiple detector rings and two-dimensional detector arrays have come to be used. The divergent x-ray geometry is called cone-beam geometry. When the multiple detector rings are close together and the rays do not spread too much, modified fan-beam type reconstruction algorithms can be used without significant image artifacts. The most often used algorithms are modifications of FBP. Other algorithms have been proposed but are too complex for efficient implementation or produce images with unacceptable artifacts.
With the above CT methods, the x-ray source rotates in a circle around the object taking fan-beam or cone-beam projections. Then the object is advanced along the axis of the system and the process repeated. In order to become even more efficient, recent systems move the object along the axis smoothly as the source continues to rotate. X-ray attenuation numbers are obtained from the two-dimensional array of detector elements as this process continues. This is called helical CT. With this geometry, the reconstruction algorithms become even more difficult. Most such algorithms work by choosing sets of source and detector locations so that approximations to fan-beams are obtained and then, for each such set, modified FBP algorithms are used.
As the geometry has gone from fan-beams to cone-beams to helical cone-beams, the need for a general and efficient reconstruction algorithm that can handle divergent beams and complex geometries has become urgent. The present invention provides a method that answers that need.